Functions

Welcome to Math Antics. In t Outside of the realm of math, the word function simply refers to what somet But in math, the word function has a more specific meaning. In math, a function is basically somet A set is just a group or collection of t Often it's a collection of n A set could be a collection of other things, Sets are sometimes shown visually But more often, you'll see sets written using a common math notation, where some or all of the members of the set are put inside curly brackets with commas between them A set can have a finite or an infinite n For example, a set containing all the letters of the alphabet has only 26 elements, while a set of all integers has an infinite n Okay, so a set is just a collection of t But how exactly does it do that? Well, to understand how functions work, it will help if we start by naming the two sets, the input set, and the output set. A function is somet And you'll often hear these input and output sets referred to by special math names. The input set is usually called the domain, and the output set is usually called the range. And it's really common to see some or all of a function's inputs and outputs listed in what we call a function table. A function table normally has two col The function itself is often written above the function table, and in the form of some sort of mathematical rule or procedure. For example, let'say that the input set of a function is a list of common polygon names, The function itself could be a simple rule that says, output the n That means if we input triangle into the function, the output will be three. And if we input square, the output will be four. If we input pentagon, the output will be five, and so on. So this function simply relates the name of a polygon to its n That's cool, but most of the functions that you'll encounter in algebra will be a little more abstract than that. They'll usually just relate one variable to another variable in the form of an equation, In this equation, if we treat x as the set of n And just For this function, we could choose any n But to keep t If we input the value 1, in other words, if we substitute the value 1 for the x in our equation, then we get y equals 2 times 1, w And since y is our output variable, we put a 2 in the output col Next, if we input the value 2 into our function, we get y equals 2 times 2, w So the output value is 4. And last, if we input the value 3 into our function, we get y equals 2 times 3, w So the output value is 6. See the pattern? For each input value, the output value is twice as big, w Okay, so we've seen some examples of functions that relate inputs to outputs, but there's an important limitation about functions that we need to know. To understand what that limitation is, let's try to make a function table for the equation y squared equals x. Again, the x variable in t Since y is our output variable, it will help if we first solve t But because of negative n But won'that mess up our function table? If we input an x value of 4, the positive or principal root would be 2. But we also have the negative root as a solution. If x equals 4, then y equals 2 and y equals negative 2 are both possible solutions to the equation y squared equals x. So in t Can a function do that? You see, functions aren't allowed to have what we call one-to-many relations, where one particular input value could result in many different output values. One-to-many relations certainly do exist, as we can see from t For somet So a function doesn't just relate a set of inputs to a set of outputs. A function relates a member of an input set to exactly one member of an output set. The equation y equals 2x qualifies as a function, because no matter what n But the equation y squared equals x does not qualify as a function, because a single input can produce more than one output. Let's look at another simple algebraic equation to see if it's a function. y equals x plus 1. Again, the x values will be inputs, the domain, and the y values will be the outputs, the range. Let's quickly generate a function table for a few possible input values, If you watched our last video about grap It's an x value followed by a y value. We could even rewrite all the inputs and outputs in ordered pair form if we wanted to. And that means you can also graph all of these pairs of inputs and outputs on the coordinate plane. You can graph a function. Here are the points from our function table plotted on the coordinate plane. And here's the resulting graph we get if we connect those points. It forms a straight line, and it's an example of what is called a linear function. In algebra, there are lots of different kinds of functions that have interesting graphs, quadratic functions, cubic functions, trig functions, and many more. These graphs may look And we can tell their functions just by looking at their graphs because they all pass the vertical line test. Remember how functions aren't allowed to have more than one output value for a particular input value? Well, the vertical line test helps us see if a graph has any of those one-to-many relations that would disqualify it as a function. Here's how it works. Imagine that a vertical line is drawn on the same coordinate plane as the graph that you want to test. Then imagine moving that vertical line left and right across the domain paying close attention to the point where the vertical line intersects with the graph. If that vertical line only intersects the graph at exactly one point for every possible value of x in the domain, then that means that there's only one output value for each input value. There's only one y value for each x value, so the graph qualifies as a function. Okay, so all of these graphs pass the vertical line test and are functions, but what's an example of a graph that doesn't pass the vertical line test? Well, here's one. It's the graph of our equation y squared equals x. The domain of t And there's one place where the vertical line would intersect the graph at just one point, w But as we move to the right on the x-axis, you can see that our vertical line is now intersecting the curve in two places. That means t Okay, now before we wrap up, we need to talk briefly about some common function notation that can be pretty confusing the first time you see it in math books. So far, we've been writing functions y equals 2x, and y equals x plus 1. But you'll often see these exact same functions written But why? Why did the variable y get replaced with that f-parentheses-x t And what does that even mean? Well, it turns out that a really common way to represent a function is t And you say it The problem with t Instead, f is the name of a function. It would be a lot more clear if mathematicians just used the entire word function as the name, and then used the names input and output instead of x and y. These two notations mean exactly the same these are the most common names, but you could use others if you wanted to. Okay, so that's the basic notation, but how did the equation get changed to f of x instead of y? Well, it comes from the idea that if two t Since we've agreed on t Either one can represent the output set of a function. But if they're interchangeable, why would you use the more complicated f of x when you could just use y instead? Well, using f of x And it gives us a handy notation for evaluating functions for specific values. For example, you could start off by saying, let the function f of x equal 3x plus 2. Then you could ask someone to evaluate the function for the input value 4 by saying, what is f of 4? That means you'll substitute a 4 in place of any x's that are in the function. For t And you could do t f of 5 equals 17, and f of 6 equals 20. Pretty easy, h Alright, so that's what functions are in math. There are t And the set of all input values is called the domain, w In algebra, functions typically come in the form of equations that can be graphed on the coordinate plane by treating the input and output values as ordered pairs. Of course, there's a lot more to learn about functions, but t Don't forget to practice using what you've learned As always, Learn more at MathAntics. com.